Activity 9.4.6

Suppose that $\vu$ and $\vv$ are vectors in $\R^n$. We wish to find $\vw_1$ and $\vw_2$ so that $\vw_1$ is parallel to $\vv$ and $\vw_2$ is orthogonal to $\vv$, as shown in this figure.

1. Explain why $\vu=\vw_1 + \vw_2$.

2. Compute $\vu\cdot \vv$ as $(\vw_1+\vw_2)\cdot \vv$. Simplify your answer as much as possible, using the fact that $\vw_2$ is orthogonal to $\vv$.

3. Since $\vw_1$ is parallel to $\vv$, there is a scalar $k$ so that $\vw_1 = k\vv$. Substitute $k\vv$ for $\vw_1$ in your answer to the previous part and then solve for $k$.

4. Give a formula for $\vw_1$ purely in terms of $\vu$ and $\vv$.